Locally-primitive block designs

One of the most important problems in combinatorial design theory is to classify flagtransitive block designs which has been extensively studied in the literature. However, the classification problem is still a widely open and challenging problem. A block design (or 𝑡-design) is a point-block incidence geometry such that each block contains a constant number of points, and any 𝑡 points lie on a constant number of blocks. In particular, for 𝑡 ⩾ 2, each t-design is also a 2-design. Observe that the flag-transitivity implies the locally-transitivity, that is, the point stabilizer is transitive on the incident blocks, and the block stabilizer is transitive on the incident points, respectively. Further, a block design is called locally-primitive if the point stabilizer is primitive on the incident blocks, and the block stabilizer is primitive on the incident points, respectively. Due to the difficulty of classifying flag-transitive designs, scholars usually focus their attention on the subclass of them. Most of the classification of flag-transitive designs focuses on flag-transitive designs with certain parameter conditions. While we limit the local action of the automorphism group to classify, that is, limiting local primitive. In group action, the 2-transitive action is also primitive, so locally 2-transitive block designs are a subclass of locally primitive designs. The main purpose of this thesis is to provide a feasible and simplified framework for the solution of this problem, which is given as follows: 1. (The Reduction Theorem) The automorphism group of a locally-primitive block design is an almost simple group or an affine group, and acts primitively on the points; 2. Give a characterization of the locally 2-transitive block designs. The proof of the above reduction theorem involves first reducing locally-primitive designs to point-primitive ones, and then analyzing the O’Nan-Scott types of primitive (quasiprimitive) groups in detail, associated with Giudici-Li-Praeger’s reduction theorem for locally-primitive graphs. Further, for the affine case in the reduction theorem, that is, 𝒟 = (𝒫, ℬ) is a locally primitive block design and the automorphism group G is an affine group, one of the following holds: (1) The action of 𝐺 on the block set ℬ is not quasiprimitive, and 𝒟 is a non-symmetric subdesign of a known design AG𝑖(𝑑, 𝑞) for some 𝑖 with 1 ⩽ 𝑖 ⩽ 𝑑 − 1. (2) 𝐺 acts primitively both on the point set 𝒫 and the block set ℬ, the design 𝒟 is symmetric, and the point stabilizer 𝐺0 is isomorphic to the block stabilizer 𝐺𝛽, but they are not necessarily conjugate in 𝐺. For the case (2), we prove that: • if 𝐺0 is an affine group, then 𝐺 ⩽ 𝐴Γ𝐿(1, 𝑝𝑑 ); • if 𝐺0 is almost simple with a sporadic socle 𝑇, then 𝑇 is 𝐽2 or 𝐶𝑜1; • if 𝐺0 is almost simple with an alternating 𝐴𝑐 , then either 5 ⩽ 𝑐 ⩽ 8; or 𝑐 ⩾ 14 and 𝒫 is isomorphic to the fully deleted permutation module for 𝐴𝑐 ; • if all complements of the socle of 𝐺 in 𝐺 have at least 2 conjugate classes, then 𝐺0 is almost simple; in particular, if 𝐺0 and 𝐺𝛽 are not conjugate in 𝐺, then 𝐺0 is almost simple. Locally 2-transitive designs constitute a special subclass of locally-primitive block designs. For a 𝐺-locally 2-transitive design 𝒟, we show that 𝐺 is also 2-transitive on the point set 𝒫, and the intersection of two blocks is empty or a constant number of points. Combined with this property, we rule out many cases by analyzing the actions of 2-transitive permutation groups in detail.

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